This is a little delayed, but I wanted to first say thank you to all here who gave me recommendations around London, and share a little about my experience. This was my first time playing golf in the UK, so I wasn't sure exactly what to expect, but ended up loving the experience, even though the weather was a bit cold and conditions a bit wet while I was there.
First: Walton Heath Old. I went here straight from Heathrow on a rainy Sunday. They insisted on charging me only half the normal rate. Everyone - the staff, members, the pros, my caddie - was incredibly welcoming and made me feel right at home.
As for the course, I loved it. It has a vastness to it (you can usually see many holes from wherever you are), and reminded me a little of Stonewall North with heather, or maybe Shinnecock (I've never played but have driven by). The sod wall bunkers were typically barely noticeable from the tee, which is probably a good thing because seeing them up close would cause me to snap hook by drive out of fear (not that I ever do that).
A few pics: